Orbit correspondences for real reductive dual pairs

Abstract

Suppose that one of the real vector spaces V and W is symplectic and the other is quadratic. Let g 1 and g 2 denote the Lie algebras of the groups of isometries of the two spaces, and let τ i : V ⊗ R W → g i be their respective moment maps for i = 1, 2. Suppose that O and L are nilpotent orbits in g 1 and g 2 , respectively. We prove that τ 2 (τ ―1 1 (O)) and τ 1 (τ ―1 2 (O))) are each the union of at most two closures of nilpotent orbits in g 1 and g 2 , respectively (where P denotes the closure of a nilpotent orbit P).